We study optomechanically induced transparency in a spinning microresonator. We find that in the presence of rotation-induced Sagnac frequency shift, both the transmission rate and the group delay of the signal are strongly affected, leading to a Fano-like spectrum of transparency. In particular, tuning the rotary speed leads to the emergence of nonreciprocal optical sidebands. This indicates a promising new way to control hybrid light–sound devices with spinning resonators.